Big Indecomposable Modules and Direct-sum Relations

The main theorem of this paper complements the tame-wild dichotomy for commutative Noetherian rings, obtained by Klingler and Levy [14]–[16]. They gave a complete classification of all finitely generated modules over Dedekind-like rings (cf. Definition 1.1) and showed that, over any ring that is not a homomorphic image of a Dedekind-like ring, the category of finite-length modules has wild representation type. A consequence of their classification is that if M is an indecomposable finitely generated module over a Dedekind-like ring R then MP is free of rank 0, 1 or 2 at each minimal prime ideal P of R. Here we prove that if (R, m, k) is a commutative local Noetherian ring of positive dimension and is not a homomorphic image of a Dedekind-like ring then there are indecomposable modules that are free of any prescribed rank at each minimal prime. This result was obtained in [9] for the case of a Cohen-Macaulay ring, using a direct but highly intricate construction. In [10] we gave a much simpler argument that handles all rings—Cohen-Macaulay or not—for which some power of the maximal ideal requires at least 3 generators. The remaining case, when (R, m, k) is not Cohen-Macaulay and each power of m is two-generated, was treated via an indirect argument using the bimodule structure of certain Ext modules. In this paper we apply the Ext argument, together with periodicity of resolutions over hypersurface rings, to give a unified treatment of the case when each power of m is two-generated. Thus this paper does not rely on the technical construction in [9]. Our goal is to make the paper pretty much self-contained, though we do refer without proof to some of the results of [6], [10] and [14]–[16]. We actually obtain max{|R/m|,א0} pairwise non-isomorphic indecomposables of each rank. This refinement allows us, in dimension one, to obtain precise defining equations for